OAI.Heisenberg.finiteRepresentation_field: intermediate lemma from OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet

Proved

OAI.Heisenberg.finiteRepresentation_field

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Heisenberg.spontaneous_magnetization, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet, OpenAI Math Release, September 24, 2026, result family 271, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Heisenberg.finiteRepresentation_field of lean/OAI/MathematicalPhysics/Heisenberg/SpinField.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Heisenberg, with open scoped Classical BigOperators Matrix Matrix.Norms.Operator, open SlotTensor TensorMatrix):

lemma finiteRepresentation_field (Λ : Finset (Site d)) (h : ℝ) :
    finiteRepresentation (ℓ:=ℓ) (fun x : Λ=>x.val) Subtype.val_injective
      ((-1/2:ℝ) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ-
        h • Matrix.diagonal (physicalMagnetization (X:=Λ) ℓ))=shiftedField Λ h

Definitions. The statement is written with OAI.Heisenberg.finiteRepresentation, OAI.Heisenberg.quasiLocalNormedAlgebra, OAI.Heisenberg.shiftedField, OAI.Heisenberg.SlotTensor.physicalMagnetization and OAI.Heisenberg.SlotTensor.spinGenerator (definitions bundle OAIW2HM000 of this split); OAI.Heisenberg.Configuration, OAI.Heisenberg.Level, OAI.Heisenberg.NearestNeighbor, OAI.Heisenberg.QuasiLocal, OAI.Heisenberg.Site, OAI.Heisenberg.SpinHilbert and OAI.Heisenberg.SpinOperator (the published bundle Def_Heisenberg).

Used by. The proof of OAI.Heisenberg.Selection.physical_field_mean of this split imports it.

Preamble

import Mathlib
import Definitions.Def_Heisenberg
import Definitions.Def_OAIW2HM000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section
noncomputable section
open scoped Classical BigOperators Matrix Matrix.Norms.Operator
namespace Heisenberg
open SlotTensor TensorMatrix
variable {d ℓ : ℕ}

lemma finiteRepresentation_field (Λ : Finset (Site d)) (h : ℝ) :
    finiteRepresentation (ℓ:=ℓ) (fun x : Λ=>x.val) Subtype.val_injective
      ((-1/2:ℝ) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ-
        h • Matrix.diagonal (physicalMagnetization (X:=Λ) ℓ))=shiftedField Λ h := by
  sorry

end Heisenberg
end
end

end OAI
end

Source

OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 271; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Heisenberg/SpinField.lean (declaration OAI.Heisenberg.finiteRepresentation_field)

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